Optimal? not

[Mike Ositoff]

  Replying to Saari's "Optimal? Not" letter:

Saari might not realize that this topic has been thoroughly
studied & described.

When I said "optimal", it should be clear that I was talking
about maximizing utility expectation. If optimality, for you,
means rating candidates differently, to distinguish them in case
of ties, then, for you, your in-between ratings are "optimal".

You said that being able to break ties is the important thing for
you, but I would point out that breaking ties is the basis for
all point-system strategy. The more likely a tie between A & B,
and the more you prefer one to the other, the more important it
is to cast that tie-breaking pair of ratings between them. And
if you chose A & B for voting a ratings difference, then why
vote a small one??

***

What's the utility gain from voting A over B in a points system?

Where P(a,b) is the probability of A & B being (the only) candidates
in a tie for winner; and where U(a) is the utility, for you, of
A, and U(b) is the utility for you of B, and where V(a) is your
vote for A, and V(b) is your vote for B; and if we assume that
the probability of A & B being exactly tied, so your vote breaks
the tie is equal to the probability of your preferred of those
2 being 1 vote behind, so you can create a tie; and if we assume
that the probability of A & B being within N votes of eachother is
N times greater than the probability of their being 1 vote apart,
then the amount by which your utility expectation is better, in
regards to candidate i than it would be if you abstained, is:

The sum, for i, over all j, of P(i,j)[U(i)-U(j)][V(i)-V(j)]

If that's summed over all i, that's how much you've improved your
utility expectation by voting instead of not showing up.

If, for a particular i & j, everything in that expression, not
counting V(i)-V(j) is positive, then that term of the sum is
maximum if V(i)-V(j) is as large as possible.

If the sum, for i, summed over all j, of P(i,j)[U(i)-U(j)]
is positive, then you're gaining more than your're losing, overall,
by maximizing all those terms, by giving maximum votes to candidate i.

So that's the strategy to maximize your utility expectation: 
If the sum in the previous paragraph is positive for i, then
give maximum votes to i. If the sum is negative for i, then
give i the minimum possible vote (even if that's negative).

***

I don't claim that that's rigorous, but hopefully it demonstrates
why it's best to give all or nothing in direct point-assignment
methods ("cardinal measure methods").

The assumptions that I stated at the beginning are made by
the people who write on the topic.

***

That sum, for i, over j, of P(i,j)[U(i)-U(j)] has been called
the "strategic value" of i.

If it's a Plurality election, you vote for the candidate with
greatest strategic value. If it's Approval you vote for 
everyone whose strategic value is positive. If it's a general
cardinal measure method, gives maximum points to everyone with
positive strategic value & minimum points to everything with
negative strategic value.

***

Mike Ossipoff